Finite element methods for semilinear elliptic stochastic partial differential equations

Abstract

We study finite element methods for semilinear stochastic partial differential equations. Error estimates are established. Numerical examples are also presented to examine our theoretical results. Mathematics Subject Classification

Sharp mean-square regularity results for SPDEs with fractional noise and optimal convergence rates for the numerical approximations

“…Nevertheless, as already discussed earlier, a more complicated form of Itô's isometry in the fBm setting makes both the regularity analysis (see the proof of Lemma 3.6) and especially the approximation error analysis (see estimates of |J 3 | 2 and |J 3 | 2 in section 4) much more involved and new techniques are needed. At last, we would like to mention two recent publications [3,4], where Cao, Hong and Liu examined strong approximations of various SPDEs driven by fractional noise with H ∈ (0, 1 2 ). Instead of the semigroup approach adopted in this paper, they used the Green function framework.…”

Error analysis of finite element approximations of the stochastic Stokes equations

“…Following the general framework of [2], we first discretize the white noise using a tensor product between Gaussian random variables and characteristic functions of the triangular elements in the triangulations of . Then we apply the standard finite element method to the stochastic Stokes equations with the white noiseẆ in (1) replaced by the discretized white noise.…”

Finite Element Method and Discontinuous Galerkin Method for Stochastic Scattering Problem of Helmholtz Type in ℝ d (d = 2, 3)

“…In [3,13], the analysis based on the traditional finite element method was successfully used on SPDEs with random coefficients, using the tensor product between the deterministic and random variable spaces. Numerical methods for SPDEs with white noise and Brownian motion added to the forcing term have also been studied [3,6,9,10].…”